Wikipedia says that Hilbert scheme of points on projective space is equivalent to the Blowup of two copies of projective space: $$(\mathbf{P}^2)^{[2]} \simeq \text{Bl}_\Delta (\mathbf{P}^2 \times \mathbf{P}^2)/S_2$$ There is also a symmetric productive of varieties (which is what I would guess as the set parameterizing two points in projective space). $$ (\mathbf{P}^{(2)})^{(2)} \simeq (\mathbf{P}^2 \times \mathbf{P}^2 ) /S^2 $$ There is no blow-up along the diagonal.
The blowup of a point in the plane has a rather involved description, as point-line combinations where $\overline{PQ} = \ell$. $$ X = \{ (Q, \ell)| P, Q \in \ell \} \subseteq \mathbf{P}^2 \times \mathbf{G}(1,2) $$ The set of points on a line are collinear in the sense of Euclidean geometry. So the blowup paramterizes the point-line combinations and by duality it can be either point-point or line-line.
$$ X = \{ [X_0:X_1:X_2] [L_0:L_1:L_2] \Big| P\cdot L = P \cdot X = 0 \} \simeq \mathbf{P^2} \times \mathbf{P^2} $$
Here the blowup is along the "exceptional divisor" $\Delta = \{ (x,x) : x \in \mathbf{P}^2 \}$.
Here are schemes with two points (length-two subschemes) that are not included in the symmetric description:
- $ \text{Spec}\big[\mathbb{C}[x,y]/(x^2, y)\big] $ (there seem to be two points here)
- $ \text{Spec}\big[\mathbb{C}[x]/(x^2) \big]$
These are possibly points in the affine Hilbert scheme rather than the rather the the projective Hilbert scheme, since we used spec.
Why are length-two subschemes in the projective plane different from finding the pairs of points in the same space? Or also, how do we picture the equivalences here?
This blowup is a particular instance of a quite general gadget called the Hilbert-Chow Morphism, because the symmetric product here is isomorphic to the Chow variety. Now the difference between Hilbert schemes and Chow varieties is that the latter parametrize effective cycles (here $0$-cycles) instead of subschemes. An effective $0$-cycle on a variety is very similar to an effective divisor on a curve: it's just a linear combination of points with multiplicity (here either a pair of distinct points, or one point counted twice; the latter case is precisely the diagonal).
The length-$2$ scheme $\operatorname{Spec}\mathbb C[x,y]/(x^2,y)$ we typically picture as a fattened version of the origin, but it's more than that; we usually draw this with an arrow pointing along $y=0$ because the intersection "remembers" that the two curves $y=x^2$ and $y=0$ were tangent in that direction (because elements of the underlying ring have linear terms in $x$, so the coefficients encode $\partial f/\partial x$). If I were to rotate both curves and get $\operatorname{Spec}\mathbb C[x,y]/(y^2,x)$, we have a scheme which we instead draw with an arrow along $x=0$ because the coefficients "remember" $\partial f/ \partial y$. This shows you that I have at least a $1$-dimensional family of distinct length-$2$ subschemes lying over the single point of the Chow variety that parametrizes the origin with multiplicity $2$.
In fact this is exactly what the fiber over a cycle of multiplicity $2$ looks like: a copy of $\mathbb P^1$ corresponding to the potential tangent directions that are forgotten when you pass from schemes to cycles. To see this, recall that the exceptional divisor of a blowup is the projectivization of the normal bundle to the center of the blowup. But the normal bundle to the diagonal $\Delta: X \to X\times X$ of any (at least smooth) variety is $N_{\Delta/X} \cong T_X$, just its tangent bundle. So the restriction of the blowup map to the exceptional divisor is $\mathbb P(T_{\mathbb P^2}) \to \mathbb P^2$, which is a $\mathbb P^1$-bundle.